Two Infinite Families Of Chiral Polytopes Of Type {4,4,4} With Solvable Automorphism Groups

We construct two infinite families of locally toroidal chiral polytopes of type {4, 4, 4}, with 1024m(2) and 2048m(2) automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2(n) where n can be any integer greater than 9. (On the other hand, no chiral polytopes of type [4, 4,4] exist for n <= 9.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21]. (C) 2020 Elsevier Inc. All rights reserved.